Deriving natural deduction rules from truth tables (Extended version)

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1 Deriving natural deduction rules from truth tables (Extended version) Herman Geuvers 1 and Tonny Hurkens 2 1 Radboud University & Technical University Eindhoven, The Netherlands herman@cs.ru.nl Abstract We develop a general method for deriving natural deduction rules from the truth table for a connective. The method applies to both constructive and classical logic. This implies we can derive constructively valid rules for any (classical) connective. We show this constructive validity by giving a general Kripke semantics, that is shown to be sound and complete for the constructive rules. For the well-known connectives, like,,, the constructive rules we derive are equivalent to the natural deduction rules we know from Gentzen and Prawitz. However, they have a different shape, because we want all our rules to have a standard format, to make it easier to define the notions of cut and to study proof reductions. In style they are close to the general elimination rules by Von Plato [14]. The rules also shed some new light on the classical connectives: e.g. the classical rules we derive for allow to prove Peirce s law. Our method also allows to derive rules for connectives that are usually not treated in natural deduction textbooks, like the if-then-else, whose truth table is clear but whose constructive deduction rules are not. We prove that if-then-else, in combination with and, is functionally complete (all other constructive connectives can be defined from it). We define the notion of cut, generally for any constructive connective and we describe the process of cut-elimination. Following the Curry- Howard isomorphism, we can give terms to deductions and we study cut-elimination as term reduction. We prove that reduction is strongly normalizing for constructive if-then-else logic. Keywords and phrases constructive logic, natural deduction, cut-elimination, Kripke semantics, Curry-Howard isomorphism Digital Object Identifier /LIPIcs... 1 Introduction Natural deduction rules come in various forms, where the tree format is the most well-known, where one either puts formulas A as the nodes and leaves of the tree, or sequents Γ A, where Γ is a sequence or a finite set of formulas. Other formalisms use a linear format, using flags or boxes to explicitly manage the open and discharged assumptions. We use a tree format with sequents in the nodes and leaves, where all rules have a special form: Γ Φ 1... Γ Φ n Γ, Ψ 1... Γ, Ψ m Γ So if the conclusion of a rule is Γ, then the hypotheses of the rule can be of one of two forms: 1. Γ, Ψ : we still need to prove D from Γ, but we are now also allowed to use Ψ as additional assumption. We call Ψ a Casus. 2. Γ Φ: in stead of proving D from Γ, we now need to prove Φ from Γ. We call Φ a Lemma. Herman Geuvers and Tonny Hurkens; licensed under Creative Commons License CC-BY Leibniz International Proceedings in Informatics Schloss Dagstuhl Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany

2 XX:2 Deriving natural deduction rules from truth tables (Extended version) One obvious advantage is that we don t have to give the Γ explicitly, as it can be retrieved from the other information in a deduction. So, we will present the deduction rules without the Γ in the format Φ 1... Φ n Ψ 1... Ψ m For every connective we have elimination rules and introduction rules, where the introduction rules come in a intuitionistic and a classical variant. The elimination rules have the following form, where we indicate occurrences of Casus by C and occurrences of Lemma by L. One of the occurrences of a Lemma is the formula we eliminate, which we indicate by E. (So Φ E below is a special case of a Lemma.) Φ E Φ L 1... Φ L n Ψ C 1... Ψ C m el The introduction rules have a classical and an intuitionistic form; the following form is the classical one, where we again indicate occurrences of Casus by C and occurrences of Lemma by L. One of the occurrences of a Casus is the formula we introduce, which we indicate by I. (So Φ I below is a special case of a Casus.) The duality between elimination and introduction is clearly visible from these rules. Φ I Φ L 1... Φ L n Ψ C 1... Ψ C m in c The intuitionistic introduction rules have the following form Φ L 1... Φ L n Ψ C 1 Φ... Ψ C m Φ in i Φ We see that, compared to the classical rule, the D has been replaced by Φ, the formula we introduce, and we have omitted the first premise, which is Φ I Φ, because it is trivial. We extract these rules from a truth table as described in the following Definition. Definition 1. Suppose we have an n-ary connective c with a truth table t c (with 2 n rows). We write ϕ = c(p 1,..., p n ), where p 1,..., p n are proposition letters and we write Φ = c(a 1,..., A n ), where A 1,..., A n are arbitrary propositions. Each row of t c gives rise to an elimination rule or an introduction rule for c in the following way. p 1... p n ϕ a 1... a n 0 p 1... p n ϕ b 1... b n 1 Φ... A j (if a j = 1) A i (if a i = 0)... el... A j (if b j = 1) A i Φ (if b i = 0)... in i Φ p 1... p n ϕ c 1... c n 1 Φ... A j (if c j = 1) A i (if c i = 0)... in c If a j = 1 in t c, then A j occurs as a Lemma in the rule; if a i = 0 in t c, then A i occurs as a Casus. The rules are given in abbreviated form and it should be understood that all judgments can be used with an extended hypotheses set Γ. So the elimination rule in full reads as follows (where Γ is a set of propositions). Γ Φ... Γ A j (if a j = 1) Γ, A i (if a i = 0)... el Γ Definition 2. Given a set of connectives C := {c 1,..., c n }, we define the intuitionistic and classical natural deduction systems for C, IPC C and CPC C as follows.

3 H. Geuvers and T. Hurkens XX:3 Both IPC C and CPC C have an axiom rule axiom( if A Γ) Γ A IPC C has the elimination rules for the connectives in C and the intuitionistic introduction rules for the connectives in C, as defined in Definition 1. CPC C has the elimination rules for the connectives in C and the classical introduction rules for the connectives in C, as defined in Definition 1. We write Γ i A if Γ A is derivable using the derivation rules of IPC C. We write Γ c A if Γ A is derivable using the derivation rules of CPC C. Example 3. From the truth table we derive the following intuitionistic rules for, 3 elimination rules and one introduction rule: A B A B -ela A B A B -elb A B A B -elc A B -in A B These rules are all intuitionistically correct, as one can observe by inspection. We will show that these are equivalent to the well-known intuitionistic rules. We will also show how these rules can be optimized and be reduced to 2 elimination rules and 1 introduction rule. From the truth table we also derive the following rules for, 1 elimination rule and 1 introduction rule, a classical and an intuitionistic one. A A -el A A -in i A A A -in c As an example of the classical derivation rules we show that A A is derivable: A, A A A, A A A, A A -el A A A, A A -in c It can be proven that A A is not derivable with the intuitionistic rules. As an example of the intuitionistic derivation rules we show that A A is derivable: A, A A A, A A -el A, A A A A -in i In the intuitionistic case, there is an obvious notion of cut that we study: an intro of Φ immediately followed by an elimination of Φ. In such case there is at least one k for which a k b k. In case a k = 0, b k = 1, we have a sub-derivation Σ of Φ k and a sub-derivation Θ of Φ k and we can plug Σ on top of Θ to obtain a derivation of. In case a k = 1, b k = 0, we have a sub-derivation Σ of Φ k Φ and a sub-derivation Θ of Φ k and we can plug Θ on top of Σ to obtain a derivation of Φ. This is then used as a hypothesis for the elimination rule (that remains in this case) in stead of the original one that was a consequence of the introduction rule (that now disappears). Note that in general there are more such k, so the general cut-elimination procedure is non-deterministic. We view this non-determinism as a natural feature in natural deduction; the fact that for some connectives (or combination of connectives), cut-elimination is deterministic is an emerging property.

4 XX:4 Deriving natural deduction rules from truth tables (Extended version) 1.1 Contribution of the paper and related work Natural deduction has been studied extensively, since the original work by Gentzen, both for classical and intuitionistic logic. Overviews can be found in [13] and [8]. Also the generalization of natural deduction to include other connectives or allow different derivation rules has been studied by various researchers. Notably, there is the work of Schroeder-Heister [11], Von Plato [14], Milne [7] and Francez and Dyckhoff [5, 4] is related to ours. Schroeder- Heister studies general formats of natural deduction where also rules may be discharged (as opposed to the normal situation where only formulas may be discharged). He also studies a general rule format for intuitionistic logic and shows that the connectives,,, are complete for it. Von Plato, Milne, Francez and Dyckhoff study generalized elimination rules, where the idea is that elimination rules arise naturally from the introduction rules, following Prawitz s [10] inversion principle: the conclusion obtained by an elimination does not state anything more than what must have already been obtained if the major premiss of the elimination was inferred by an introduction. A difference is that we focus not so much on the rules but on the fact that we can define different and new connectives constructively. In our work, we do not take the introduction rules as primary, with the elimination rules defined from them, but we derive elimination and introduction rules directly from the truth table. Then we optimize them, which can be done in various ways, where we adhere to a fixed format for the rules. Many of the generalized elimination rules, for example for, appear naturally as a consequence of our approach. of deriving the rules from the truth table. The idea of deriving deduction rules from the truth table also occurs in the work of Milne [7], but in a slightly different way: from the introduction rules, a truth table is derived and then the classical elimination rules are derived from the truth table. For the if-then-else connective, this amounts to rules equivalent to ours (see Appendix 8), but not optimized. We start from the truth table and also derive rules for constructive logic. In Section 3 we give a complete Kripke semantics for the constructive connectives. This is reminiscent of the Lindenbaum lemma that is used in [7] to prove classical completeness. The Kripke semantics also allows us to prove some meta properties about the rules. For example, we give a generalization of the disjunction property in intuitionistic logic. In Section 4 we define and study cuts precisely, for the intuitionistic case. We look more in detail into the logic with just if-then-else and we prove that cut-elimination is strongly normalizing by studying the reduction of proof terms. 2 Simple properties and examples We first define precisely how the plugging one derivation in another works. Lemma 4. If Γ ϕ and, ϕ ψ, then Γ, ψ Proof. By a simple induction on the derivation of, ϕ ψ, using the fact that, in general (for all Γ, Γ and ϕ): If Γ ϕ and Γ Γ, then Γ ϕ. We can be a bit more precise about what is happening in the proof of Lemma 4. If Π is the derivation of, ϕ ψ, due to the format of our rules, the only place in Π where the hypothesis ϕ can be used is at a leaf of Π, in an instance of the (axiom) rule. These leaves are of the shape, ϕ ϕ for some. If Σ is the derivation of Γ ϕ, then Σ is also a derivation of, Γ ϕ (for any ). So, we can replace each leaf of Π that is an instance of an axiom, ϕ ϕ by a derivation Σ of, Γ ϕ, to obtain a derivation of Γ, ψ. We introduce some notation to support this.

5 H. Geuvers and T. Hurkens XX:5 Notation 5. If Σ is a derivation of Γ ϕ and Π is a derivation of, ϕ ψ, then we have a derivation of Γ, ψ that looks like this: Γ ϕ... Π ψ Γ ϕ So in Π, every application of an (axiom) rule at a leaf, deriving ϕ for some is replaced by a copy of a derivation Σ, which is also a derivation of, Γ ϕ. In Definitions 1 and 2, we have given the precise rules for our logic, in intuitionistic and classical format. We can freely reuse formulas and weaken the context, so the structural rules of contraction and weakening are wired into the system. In examples, to simplify derivations we will often use the following format for an elimination rule (and equivalently for an introduction rule). Γ 0 Φ... Γ j A j (if a j = 1) Γ i, A i (if a i = 0)... el n k=0γ k To reduce the number of rules, we can take a number of rules together and drop one or more hypotheses. We show this by again looking at the example of the rules for (Example 3). Example 6. From the truth table we have derived the following 3 intuitionistic elimination rules: A B A B -ela A B A B -elb These rules can be reduced to the following 2 equivalent elimination rules: A B A -el1 A B B -el2 A B A B -elc It can be shown that these sets of rules are equivalent (and equivalent again to the more standard -elimination rules that are known as first and second projection ). Here we only show the derivability of the new rules from the standard ones (the ones we have derived from the truth table), by giving a derivation for the -el 1 rule. (The -el 2 rule is similar.) Suppose we have derivations of Γ A B and of Γ, A. Then we have the following derivation, using the rules -el a and -el b : Γ A B Γ, A Γ, B A B Γ, B, A Γ, B B Γ Γ, B The general method here is that we can replace two rules that only differ in one hypothesis, which in one rule occurs as a Lemma and in the other as a Casus, by one rule where the hypothesis is removed. It will be clear that the Γ s above are not relevant for the argument, so we will not write these. Lemma 7. A system with two derivation rules of the form Φ 1... Φ n Ψ 1... Ψ m A Φ 1... Φ n A Ψ 1... Ψ m

6 XX:6 Deriving natural deduction rules from truth tables (Extended version) is equivalent to the system with these two rules replaced by Φ 1... Φ n Ψ 1... Ψ m Proof. The implication from bottom to top is immediate. From top to bottom, suppose we have the two given rules. We now derive the bottom one. Assume we have derivations of Φ 1,..., Φ n, Ψ 1,..., Ψ m. We now have the following derivation of. Φ 1... Φ n Ψ 1... Ψ m A Φ 1... A Φ n A A A, Ψ 1... A, Ψ m A Similarly, we can replace a rule which has only one Casus by a rule where the Casus is the conclusion. To illustrate this: the simplified elimination rules for, -el 1 and -el 2 have only one Casus. The rule -el 1 (left) can thus be replaced by the rule -el 1 (right), which is the usual projection rule. A B A -el1 A B -el 1 A There is a general Lemma stating this simplification is correct. Lemma 8. A system with a derivation rule of the form to the left is equivalent to the system with this rule replaced by the rule on the right. Φ 1... Φ n Ψ Φ 1... Φ n Proof. The implication from left to right is immediate. From right to left, assume we have derivations of Φ 1,..., Φ n. Then, by the rule to the right, we have Γ Ψ. Now assume we also have a derivation of Ψ. By Lemma 4, we also have a derivation of Γ. 2.1 The connective if-then-else If we look at if-then-else, which has the obvious (classical) truth table semantics as a ternary connective (see also the Appendix 8), and we apply the optimizations of Lemmas 7 and 8 we obtain the following intuitionistic rules, where we write A B/C for if A then B else C. A B/C B A then-el A B then-in A B/C Ψ A B/C A C else-el A A B/C A B/C C else-in Basically, A B/C is equivalent to (A B) (A C). It can be shown that A B/C is in between other constructive renderings of if-then-else: (A B) ( A C) A B/C (A B) ( A C) The left-to-right can easily be derived, for the non-derivability of the reverse, we need a Kripke model (see Section 3). If we compare with well-known classical rules for if-then-else, we observe that two of them hold, while the other fails.

7 H. Geuvers and T. Hurkens XX:7 Fact if A then B else B B and B if A then B else B, 2. if (if A then B else C) then D else E if A then (if B then D else E) else (if C then D else E) 3. if A then (if B then D else E) else (if C then D else E) if (if A then B else C) then D else E. As a matter of fact, either one of the last two rules renders the connective if-then-else classical. This can be observed by taking in (2) B =, C =, D =, E =. Then the left-hand-side is equivalent with A and the right-hand-side is equivalent with A. In (3), take B =, C =, D =, E = A. Then the left-hand-side is equivalent with and the right-hand-side with A A. So, the addition of either one of these judgments as a rule renders the system classical. An important property is that (just as in classical logic), the constructive if-then-else, together with and is functionally complete: all other connectives can be defined in terms of it. We prove this for,, and. A result from Schroeder-Heister [11] implies that all constructive connectives can be defined in terms of if-then-else. Definition 10. The derivation rules for the standard intuitionistic connectives are the following. These rules are derived from the truth tables and optimized following Lemmas 7 and 8. The rules for are the intro rule of Example 3 and the elimination rules of Example 6. The rules for are given in Example 3. The rules for and and and are: A B A B -el A B B A -el A -in1 A B B -in1 A B B -in2 A B A A B -in2 A B -in -el Example 11. As our only example for classical logic, we give the classical rules for implication. The elimination rule is the same, -el above, and we also have the first introduction rule -in 1, but in addition we have the rule -in c 2. We observe that this rule is classical in the sense that one can derive Peirce s law, without using negation. See the derivation below, of Peirce s law. A A B -in c 2 A A A ((A B) A) A (A B) A (A B) A A B A B -el A B, (A B) A A -in 1 A B, (A B) A ((A B) A) A -in2 ((A B) A) A A B ((A B) A) A -in c 2 Definition 12. We define the usual intuitionistic connectives in terms of if-then-else, and as follows. A B := A A/B A B := A B/A A B := A B/ A := A / The following is now a routine check. Lemma 13. The defined connectives in Definition 12 satisfy the derivation rules for these same connectives as given in Definition 10 Corollary 14. The intuitionistic connective if-then-else, together with and, is functionally complete. Proof. Lemma 10 shows that the well-known intuitionistic connectives can all be defined in terms of if-then-else, and. In [11], it is shown that all connectives can be defined in terms of,, and.

8 XX:8 Deriving natural deduction rules from truth tables (Extended version) 3 Kripke semantics We now define a Kripke semantics for the intuitionistic rules and prove that it is complete. We follow standard methods, given e.g. in [12, 13], which we generalize to arbitrary connectives. The concrete model we construct to prove completeness is also reminiscent of the Lindenbaum s lemma that is used in [7] to prove completeness for a classical semantics. Formulas are built from atoms using existing or defined connectives of any arity, so for each n-ary connective c, we assume a truth table t c : {0, 1} n {0, 1} and we have inductively defined derivability as a relation between a sets of formulas and a formula above. Definition 15. We define a Kripke model as a triple (W,, at) where W is a set of worlds with a reflexive, transitive relation on it and a function at : W (At) satisfying w w at(w) at(w ). In a Kripke model we want to define the relation w ϕ between worlds and formulas (ϕ is true in world w). We do this by defining [[ϕ] w {0, 1}, with the meaning that [[ϕ] w = 1 if w ϕ and [[ϕ] w = 0 if w ϕ. Definition 16. Given a Kripke model (W,, at) we define [[ϕ] w {0, 1}, by induction on ϕ as follows. (atom) if ϕ is atomic, [[ϕ] w = 1 iff ϕ at(w). (connective) for ϕ = c(ϕ 1,..., ϕ n ), [[ϕ] w = 1 iff for each w w, t c ([[ϕ 1 ] w,..., [[ϕ 1 ] w ) = 1 where t c is the truth table of c. We define Γ = ψ (ψ is a consequence of Γ) as: for each Kripke model and each world w, if for each ϕ in Γ, [[ϕ] w = 1, then [[ψ ] w = 1. Lemma 17 (Soundness). If Γ ψ, then Γ = ψ Proof. By induction on the derivation of Γ ψ. Now we prove completeness: if Γ = ψ, then Γ ψ. We prove this by constructing a special, universal Kripke model. Definition 18. For ψ a formula and Γ a set of formulas, we say that Γ is a ψ-maximal set of formulas if Γ ψ and for every formula ϕ / Γ we have: Γ, ϕ ψ. Given a formula ψ and a set of formulas Γ such that Γ ψ, we can extend Γ to a ψ-maximal set Γ that contains Γ as follows. Take an enumeration of the formulas as ϕ 1, ϕ 2,... and define recursively Γ 0 := Γ and Γ i+1 := Γ i if Γ i, ϕ i+1 ψ and Γ i+1 := Γ i, ϕ i+1 if Γ i, ϕ i+1 ψ. Then take Γ := i N Γ i. (NB. as always, Γ i, ϕ i+1 denotes Γ i {ϕ i+1 }.) Fact 19. We list a couple of simple important facts about ψ-maximal sets Γ. 1. For every ϕ, we have ϕ Γ or Γ, ϕ ψ. 2. So, for every ϕ, if ϕ / Γ then Γ, ϕ ψ. 3. For every ϕ, if Γ ϕ, then ϕ Γ. (This follows by Lemma 4, taking = Γ. If ϕ / Γ, then Γ, ϕ ψ which together with Γ ϕ yields Γ ψ). Definition 20. We define the Kripke model U = (W,, at) as follows: A world w W is a pair (Γ, ψ) where Γ is a ψ-maximal set of formulas. (Γ, ψ) (Γ, ψ ) := Γ Γ. at(γ, ψ) := Γ At. Lemma 21. In the model U we have, for all worlds (Γ, ψ) W : ϕ, ϕ Γ [[ϕ] (Γ,ψ) = 1.

9 H. Geuvers and T. Hurkens XX:9 Proof. The proof is by induction on ϕ. If ϕ At, the result is immediate, so suppose that ϕ = c(ϕ 1,..., ϕ n ) where c has truth table t c. We prove the two directions separately. ( ): Assume ϕ Γ. We have [[ϕ]] (Γ,ψ) = 1 iff for all Γ Γ and for all ψ, writing w = (Γ, ψ ), we have t c ([[ϕ 1 ] w,..., [[ϕ n ] w ) = 1. So let Γ Γ and let ψ be a formula such that Γ is ψ -maximal. For the sub-formulas of ϕ we have the following possibilities [[ϕ j ] w = 1, and then by induction hypothesis: ϕ j Γ and so Γ ϕ j. [[ϕ i ] w = 0, and then by induction hypothesis: ϕ i / Γ and so Γ, ϕ i ψ. This corresponds to an entry in the truth table t c for the connective c. Suppose t c ([[ϕ 1 ] w,..., [[ϕ n ] w ) = 0. Then this row in the truth table yields an elimination rule that allows us to prove ψ : Γ ϕ... Γ ϕ j (for ϕ j with [[ϕ j]] w = 1) Γ, ϕ i ψ (for ϕ i with [[ϕ i]] w = 0)... el Γ ψ Note that all hypotheses of the rule are derivable, because ϕ Γ and the other hypotheses are derivable by induction. So we have Γ ψ. Contradiction! So: t c ([[ϕ 1 ] w,..., [[ϕ n ] w ) = 1, what we needed to prove. ( ): Assume [[ϕ] (Γ,ψ) = 1 and suppose (towards a contradiction) ϕ / Γ. Then Γ ϕ (because if Γ ϕ, then ϕ Γ by the facts we remarked about Kripke model U.) So there is a ϕ-maximal theory Γ Γ with Γ ϕ. So (Γ, ϕ) is a world in U with (Γ, ψ) (Γ, ϕ). We write w := (Γ, ϕ) and we have t c ([[ϕ 1 ] w,..., [[ϕ n ] w ) = 1. We consider the different sub-formulas of ϕ: the ϕ j with [[ϕ j ] w = 1, and so (by induction hypothesis) ϕ j Γ and so Γ ϕ j ; the ϕ i with [[ϕ i ] w = 0, and so (by induction hypothesis) ϕ i / Γ and so Γ, ϕ i ϕ. Now, using an introduction rule for connective c, we can derive ϕ: Γ ϕ j (for ϕ j with [[ϕ j]] w = 1) Γ, ϕ i ϕ (for ϕ i with [[ϕ i]] w = 0)... in Γ ϕ So we have Γ ϕ, because the hypotheses of the rule are all derivable. Contradiction! So ϕ Γ. Theorem 22. If Γ = ψ, then Γ ψ. Proof. Suppose Γ = ψ and Γ ψ. Then we can find a ψ-maximal superset Γ of Γ such that Γ ψ. In particular: ψ is not in Γ. So (Γ, ψ) is a world in the Kripke model U in which each member of Γ is true: [[ϕ] (Γ,ψ) = 1 for all ϕ Γ, by Lemma 21. However, ψ is not true in (Γ, ψ): [[ψ ] (Γ,ψ) = 0. So Γ = ψ. Contradiction, so Γ ψ. In intuitionistic logic, the disjunction connective has a special property that does not hold for classical logic, called the disjunction property: If A B, then A or B. This implies that the disjunction is strong : if one has a proof of a disjunction, one has a proof of one of the disjoints. (Which is classically not the case, viz. A A.) The disjunction property can easily be proved using Kripke semantics, relying on the completeness theorem. We want to generalize this to our new connectives and we introduce the notion of a splitting connective.

10 XX:10 Deriving natural deduction rules from truth tables (Extended version) Definition 23. Let c be an n-ary connective, 1 i, j n. We say that c is i, j-splitting in case the truth table for c has the following shape p 1... p i... p j... p n c(p 1,..., p n) So, in all rows where p i = p j = 0 we have c(p 1,..., p n ) = 0. Phrased purely in terms of t c, that is: t c (p 1,..., p i 1, 0, p i+1,..., p j 1, 0, p j+1,..., p n ) = 0 for all p 1,..., p i 1, p i+1,..., p j 1, p j+1,..., p n {0, 1}. Note that a connective can be i, j-splitting for more than one i, j-pair. Examples are the ternary connectives most and if-then-else. See Appendix 8. In [12] and [13], the completeness of Kripke semantics is proved using prime theories (which in [12] are called saturated theories.) A theory is a set of formulas Γ that is closed under and a prime theory is defined as a theory that satisfies the disjunction property: if Γ A B, then Γ A or Γ B. (This is equivalent to A B Γ implies A Γ or B Γ.) We generalize the disjunction property to arbitrary n-ary intuitionistic connectives. Definition 24. A theory is a set of formulas Γ that is closed under. We say that Γ is a prime theory if for all i, j-splitting connectives c, in case c(a 1,..., A n ) Γ, then A i Γ or A j Γ. Lemma 25. If Γ is ψ-maximal then Γ is a prime theory. Proof. Let Γ be a ψ-maximal set of formulas. Obviously, Γ is closed under derivability, so it is a theory. Let c be an i, j-splitting connective and let ϕ = c(a 1,..., A n ) Γ. Suppose A i / Γ and A j / Γ. Because Γ is ψ-maximal, this means that Γ, A i ψ and Γ, A j ψ. For the other formulas A k {A 1,..., A i 1, A i+1,..., A j, A j+1,..., A n } we do not know whether A k Γ (and then Γ A k ) or A k / Γ (and then Γ, A k ψ), but for each A k either one of the two is the case. Because c is i, j-splitting, we have an elimination rule. Γ ϕ... Γ, A i ϕ... Γ, A j ϕ... el Γ ψ All hypotheses are derivable, so the conclusion is derivable. Contradiction! So A i Γ or A j Γ. We now prove our generalization of the disjunction property. Lemma 26. Let c be an i, j-splitting connective and suppose c(a 1,..., A n ). Then A i or A j. Proof. Let c be an i, j-splitting connective and let ϕ = c(a 1,..., A n ) be a formula with ϕ. Suppose A i and A j. Then there are Kripke models K 1 and K 2 such that K 1 A i and K 2 A j. We may assume that the sets of worlds of K 1 and K 2 are disjoint so we can construct a Kripke model K as the union of K 1 and K 2 where we add a special root world w 0 that is below all worlds of K 1 and K 2, with at(w 0 ) =. It is easily verified that K is a Kripke model and we have w 0 A i, because w 0 is below some world w in K 1 with w A i ; similarly w 0 A j. So, [[A i ]] w0 = [[A j ]] w0 = 0. But then w 0 ϕ, because

11 H. Geuvers and T. Hurkens XX:11 [[ϕ]] w0 = [[c(a 1,..., A n )]] w0 = 1 iff for all w w 0 : t c ([[A 1 ]] w,..., [[A n ]] w ) = 1. However, for w := w 0, whatever the values of [[A k ]] w are for k i, j, t c ([[A 1 ]] w,..., [[A n ]] w ) = 0. On the other hand, w 0 ϕ, because ϕ, so we have a contradiction. We conclude that A i or A j. 4 Cuts and cut-elimination The idea of a cut in intuitionistic logic is an introduction of a formula Φ immediately followed by an elimination of Φ. We will call this a direct intuitionistic cut. In general in between the intro rule for Φ and the elim rule for Φ, there may be other auxiliary rules, so occasionally we may have to first permute the elim rule with these auxiliary rules to obtain a direct cut that can be contracted. We leave that for future research and now just define the notion of direct cut and contraction of a direct cut. Definition 27. Let c be a connective of arity n, with an elim rule and an intuitionistic intro rule derived from the truth table, as in Definition 1. So suppose we have the following rules in the truth table t c. p 1... p n c(p 1,..., p n) a 1... a n 0 b 1... b n 1 An intuitionistic direct cut in a derivation is a pattern of the following form, where Φ = c(a 1,..., A n ).... j Γ A j i Γ, A i Φ... Γ Φ... Πk Γ A k Γ Πl Γ, A l... A j ranges over all propositions where b j = 1; A i ranges over all propositions where b i = 0, A k ranges over all propositions where a k = 1; A l over all propositions where a l = 0, The elimination of a direct cut is defined by replacing the derivation pattern above by 1. If l = j (for some l, j): j j Γ A j... Γ A j Πl Γ 2. If k = i (for some k, i): Πk Πk Γ A i... Γ A i i Γ Φ... Πk Γ A i Γ Πl Γ, A l... There may be several choices for the i and j in the previous definition, so cut-elimination is non-deterministic in general. We study the example of if-then-else in more detail.

12 XX:12 Deriving natural deduction rules from truth tables (Extended version) Example 28. The intuitionistic cut-elimination rules for if-then-else are the following. (then-then) Γ A Γ B Γ A B/C Γ A Γ B Γ B (else-then) Γ, A A B/C Γ C Γ A B/C Γ B Π Γ A Π Π Γ A... Γ A Γ A B/C Γ B Π Γ A (then-else) Γ A Γ B Γ A B/C Π Γ, A Γ, C Γ Γ A... Π Γ Γ A (else-else) Γ, A A B/C Γ A B/C Γ C Γ, A Γ Π Γ, C Γ C... Π Γ Γ C 5 A Curry-Howard isomorphism We now define types terms for derivations, which enables the study of proofs as terms and emphasizes the computational interpretation of proofs. Here, we only define terms for derivations in the inituitionistic logic, which can be extended to the classical logic in an obvious way. We first define terms associated with connectives in general. Then, to show the usefulness of our approach to logical connectives, we will focus on the if-then-else connective. Definition 29. Suppose we have a logic with intuitionistic derivation rules, as derived from truth tables for a set of connectives C, as in Definition 1. We define rules for the judgment Γ t : A, where A is a formula, Γ is a set of declarations {x 1 : A 1,..., x m : A m }, where the A i are formulas and the x i are term-variables such that every x i occurs at most once in Γ, and t is a proof-term as follows. For every connective c C of arity n, we have an introduction term ι(t 1,..., t n ) and an elimination term ε(t 0, t 1,..., t n ), where the t i are again terms of the shape λx.t, where x is a term-variable and t is a term. The terms are typed using the following derivation rules. x i : A i Γ Γ x i : A i... Γ t j : A j Γ, y i : A i q i : Φ... in Γ ι( t, λy.q) : Φ

13 H. Geuvers and T. Hurkens XX:13 Γ t 0 : Φ Γ t k : A k Γ, y l : A l q l : D el Γ ε(t 0, t, λy.q) : D Here, t is the sequence of terms t 1,..., t p for all the 1-entries in the truth table, and λy.q is the sequence of terms λy 1.q 1,..., λy r.q r for all the 0-entries in the truth table. One may think of the λ-abstracted variables as being typed so then one could write λy : A.q and λy 1 : A 1.q 1,..., λy r : A r.q r. However, this clutters up the syntax considerably, so we will leave these types implicit. Moreover, decidability of typing, or a typing algorithm for (untyped) terms of the calculus is not our concern here. There are term reduction rules that correspond to the elimination of direct cuts. Definition 30. Given a direct cut as defined in Definition 27, we add reduction rules for the associated terms as follows. (For simplicity of presentation we write the matching cases in Definition 27 as last term of the sequence.) For the l = j case: ε(ι( t, t j, λy.q), s, λy.r, λy l.r l ) r l [y l := t j ] For the k = i case: ε(ι( t, λy.q, λy i.q i ), s, s k, λy.r) ε(q i [y i := s k ], s, s k, λy.r) The reduction is extended in the straightforward way to sub-terms, by defining it as a congruence with respect to the term constructions. This Definition gives a reduction rule for every combination of an elimination and an introduction. For an n-ary connective, there are 2 n rules in the truth table, and therefore 2 n constructors (introduction plus elimination constructors). Often, we will want to just look at the optimized rules, following Lemmas 7 and 8. For these optimized rules, there is also a straightforward definition of proof-terms and of the reduction relation associated with cut-elimination. The Lemmas 7 and 8 can be extended to terms and reductions: the proof-terms for the optimized rules can be defined in terms of the terms for the original calculus, and the reduction rules for the optimized proof terms are an instance of reductions in the original calculus (often multi-step). We now focus on the logic with just if-then-else. We define a calculus λif-then-else for proof terms for this logic. Definition 31. We define the calculus λif-then-else as a calculus for terms and reductions for the if-then-else logic as follows. (To understand the reduction rules, also look at Section 2.1.) Γ t 0 : A B/C Γ ε 1(t 0, a) : B Γ a : A then-el Γ a : A Γ b : B then-in Γ ι 1(a, b) : A B/C The reduction rules are Γ t 0 : A B/C Γ, x : A t : D Γ, y : C q : D else-el Γ ε 2(t 0, λx.t, λy.q) : D Γ, x : A t : A B/C Γ c : C else-in Γ ι 2(λx.t, c) : A B/C ε 1 (ι 1 (a, b), a ) b ε 1 (ι 2 (λx.t, c), a) ε 1 (t[x := a], a) ε 2 (ι 1 (a, b), λx.t, λy.q) t[x := a] ε 2 (ι 2 (λx.t, c), λz.d, λy.q) q[y := c] Here t[x := a] denotes the substitution of a for x in t.

14 XX:14 Deriving natural deduction rules from truth tables (Extended version) The interpretation of intuitionistic proposition logic can be extended to include proof terms. These are well-known for proposition logic. e.g. see [13]. The interpretation of Definition 12 can be extended to the proof terms. If we denote this translation by a ( ), we find that, if Γ t : A in intuitionistic proposition logic with proof terms, then Γ ṫ : A in the calculus λif-then-else of Definition 31. Lemma 13 also extend to term reductions: If t q for the proof terms t and q in intuitionistic proposition logic, then ṫ + q in λif-then-else (where + denotes the transitive closure of ). We do not define the ( ) function precisely, nor do we prove the mentioned properties about it here. The reason is that we merely want to use it as one of the motivations for studying strong normalization of reduction of proof terms of λif-then-else. Strong normalization is the property that a term has no infinite reduction starting from it. Strong normalization for λif-then-else implies (using the properties of the ( ) function) strong normalization of proof term reduction in intuitionistic proposition logic. 5.1 Strong Normalization We prove SN by adapting the well-known saturated sets method of Tait to our calculus. We write SN for the set of strongly normalizing (untyped) terms and we write Term for the set of all untyped terms and Var for the set of variables. Definition The set Neut of neutral terms is defined by (a) Var Neut, (b) ε 1 (t 0, a) Neut for all t 0 Neut and a SN, (c) ε 2 (t 0, λx.t, λy.q) Neut for all t 0 Neut, t, q SN. 2. The term t does a key reduction to q, notation t k q, in case (i) t is a redex itself (according to Definition 31) and q is its reduct, or (ii) t = ε 1 (t 0, a), q = ε 1 (q 0, a) and t 0 k q 0, or (iii) t = ε 2 (t 0, λx.r, λy.s), q = ε 2 (q 0, λx.r, λy.s) and t 0 k q A set X Term is saturated if it satisfies the following properties (i) X SN, (ii) Neut X and (iii) X is closed under key-redex expansion: if q X, t SN and t k q, then t X. 4. Given X, Y, Z SAT we define the set X Y/Z by X Y/Z := {M a X(ε 1 (M, a) Y ) D SAT, t, q Term, a X(t[x := a] D) c Z(q[y := c] D) = ε 2 (M, λx.t, λy.q) D } Lemma 33. If X, Y, Z SAT, then X Y/Z SAT. See Appendix 8 for a proof. We use the saturated sets as a semantics for types: if A is a type, A will be a saturated set. The simplest way to do this is to interpret all type variables (proposition letters) as the set SN (which is indeed a saturated set) and interpret A B/C as A B / C, where this definition is from Definition 32. Definition 34. Given a context Γ, a map (valuation) ρ : Var Term satisfies Γ, notation ρ = Γ, in case ρ(x) A for all x : A Γ. If t Term and ρ : Var Term, we write t ρ for t where ρ has been carried out as a substitution on t. A valuation ρ : Var Term is only relevant for a finite number of variables: those that are declared in the context Γ under consideration. So we will always assume that ρ(x) x only for a finite number of x Var. Those x we call the support of ρ. When applying ρ as a

15 H. Geuvers and T. Hurkens XX:15 substitution to a term t we may need to go under a λ, e.g. when applying ρ to ι 2 (λx.t, c) In this case we always assume that the bound variable is not in the support of ρ. (We can always rename it.) This allows us to just write ι 2 (λx.t, c) ρ = ι 2 (λx. t ρ, c ρ ). Lemma 35. If Γ t : A, and ρ = Γ, then t ρ A. For a proof, see Appendix 8 The following is now immediate by taking ρ(x) := x for all x Var. Because Var Neut A, we know that ρ = Γ. So, if Γ t : A, then t ρ = t A SN. Corollary 36. The system λif-then-else is strongly normalizing: all reductions on proof terms are finite. 6 Conclusion and Further work We have introduced a general procedure for deriving natural deduction rules from truth tables that applies both to classical and intuitionistic logic. Our deduction rules obey a specific format, making it easier to study. To show that the intuitionistic rules are truly constructive we have defined a complete Kripke semantics for the intuitionistic rules. We have defined cut-elimination for intuitionistic logic in general and given a Curry-Howard proofs-as-terms isomorphism for it. We have studied it in more detail for if-then-else and proved the reduction on proof-terms to be strongly normalizing. The work described here raises very many new research questions: Is cut-elimination normalizing in general for an arbitrary set of connectives? How to define cut-elimination for the classical case, and what is its connection with a term calculus, for example calculi for classical logic studied in [1] and [9]? Also [3] defines a whole series of systems and connectives, of which the of [2] is just one. How do the computation rules compare with ours? On a more technical note: In λif-then-else, a cut can be hidden in case the second or third sub-derivation of an else-elrule ends with an introduction of a formula D := E F/G which is eliminated after the else-el. The idea is to permute the elimination over the application of the else-elrule. This can be achieved by the following permuting reduction rules: ε 1 (ε 2 (t 0, λx.t, λy.q), e) ε 2 (t 0, λx.ε 1 (t, e), λy.ε 1 (q, e)) ε 2 (ε 2 (t 0, λx.t, λy.q), λv.r, λz.s) ε 2 (t 0, λx.ε 2 (t, λv.r, λz.s), λy.ε 2 (q, λv.r, λz.s)) These reduction rules correspond to obvious transformations of derivations, permuting one elimination over another. The normal forms t for this combined reduction are such that all sub-terms of t have types that are sub-types of the type of t or sub-types of types of free variables in t. We leave it for future research to prove that λif-then-else with these permuting reductions is normalizing. Techniques as in [6], where a similar property is proved for intuitionistic logic with permuting cuts, may be useful. 7 References References 1 Z. Ariola and H. Herbelin. Minimal classical logic and control operators. In ICALP, volume 2719 of LNCS, pages Springer, Tristan Crolard. A formulae-as-types interpretation of subtractive logic. J. Log. Comput., 14(4): , 2004.

16 XX:16 Deriving natural deduction rules from truth tables (Extended version) 3 P.-L. Curien and H. Herbelin. The duality of computation. In ICFP, pages , Roy Dyckhoff. Some remarks on proof-theoretic semantics. In Advances in Proof-Theoretic Semantics, pages Springer, N. Francez and R. Dyckhoff. A note on harmony. Journal of Philosophical Logic, 41(3): , F. Joachimski and R. Matthes. Short proofs of normalization for the simply- typed lambdacalculus, permutative conversions and Gödel s T. Arch. Math. Log., 42(1):59 87, Peter Milne. Inversion principles and introduction rules. In Dag Prawitz on Proofs and Meaning, Outstanding Contributions to Logic, vol 7, pages Springer, S. Negri and J. von Plato. Structural Proof Theory. Cambridge University Press, M. Parigot. λµ-calculus: An algorithmic interpretation of classical natural deduction. In LPAR, volume 624 of LNCS, pages Springer, Dag Prawitz. Ideas and results in proof theory. In J. Fenstad, editor, 2nd Scandinavian Logic Symposium, pages North-Holland, P. Schroeder-Heister. A natural extension of natural deduction. J. Symb. Log., 49(4): , A.S. Troelstra and D. van Dalen. Constructivism in Mathematics Vol.1. Elsevier, Dirk van Dalen. Logic and structure (3. ed.). Universitext. Springer, Jan von Plato. Natural deduction with general elimination rules. Arch. Math. Log., 40(7): , Appendix Truth tables of most and if-then-else A B C most(a, B, C) A B/C We see that most is i, j-splitting for every i, j. Indeed, if most(a, B, C), we can derive A or B but also A or C and also B or C. The connective if-then-else is not 1, 2-splitting but it is 1, 3-splitting and 2, 3-splitting: if A B/C, then we have A or C and also B or C. From the lines in the truth table of A B/C with a 0 we get the following four elimination rules: A B/C A B C A B/C A B C A B/C A B C A B/C A B C Using Lemmas 7 and 8, these can be reduced to the following two. (The two rules on the first line reduce to else-el, the two rules on the second line reduce to then-el.) A B/C A C else-el A B/C B A then-el

17 H. Geuvers and T. Hurkens XX:17 From the lines in the truth table of A B/C with a 1 we get the following four classical introduction rules: A B/C A B C A B/C A B C A B/C A B C A B/C A B C Using Lemmas 7 and 8 can be reduced to the following two. (The two rules on the first line reduce to else-in, the two rules on the second line reduce to then-in.) A B/C A C else-in c A B then-in A B/C These are the classical rules for if-then-else. Only the last rule, else-in c, is different from the constructive one, as given in Example 2.1. Truth tables and rules for subtraction and bi-implication We now treat subtraction of [2] and bi-implication. The classical reading of A B is A B, so we have the truth table below. A B A B A B This yields the following intuitionistic derivation rules. For subtraction, these are the same as [2]. A B A B in A B A el1 A B B el2 A B A A B B in1 A A B A B B A B in2 A B A B el1 A B A B el2 Proof of Lemma 33 If X, Y, Z SAT, then X Y/Z SAT. Proof. We check the 3 conditions for X Y/Z SAT. (i) X Y/Z SN follows directly from the fact that if M X Y/Z, then ε 1 (M, x) Y and Y SN, so ε 1 (M, x) SN, so M SN. (ii) We check the inductive cases for Neut:

18 XX:18 Deriving natural deduction rules from truth tables (Extended version) (a) Var X Y/Z because for z Var and a X, ε 1 (z, a) Neut Y and for D SAT and t, q SN, ε 2 (z, λx.t, λy.q) Neut D. (b) if M = ε 1 (t 0, a), then for all a X, ε 1 (ε 1 (t 0, a), a ) Neut Y and for all D SAT and t, q SN, we have ε 2 (ε 1 (t 0, a), λx.t, λy.q) Neut D. (c) if M = ε 2 (t 0, λx.t, λy.q) then for all a X, ε 1 (ε 2 (t 0, a), λx.t, λy.q), a ) Neut Y. For D SAT and t, q SN we have ε 2 (ε 2 (t 0, λx.t, λy.q), λx.t, λy.q ) Neut D. (iii) Suppose M k M with M X Y/Z. Then for all a X, ε 1 (M, a) k ε 1 (M, a) and ε 1 (M, a) Y, so ε 1 (M, a). Similarly, M satisfies the second condition in the definition of X Y/Z, so M X Y/Z. Proof of Lemma 35 If Γ t : A, and ρ = Γ, then t ρ A. Proof. By induction on the derivation of Γ t : A. Suppose ρ = Γ. For the (axiom) case, it is trivial. Suppose Γ t 0 : A B/C Γ a : A then-el Γ ε 1 (t 0, a) : B Then ε 1 (t 0, a) ρ = ε 1 ( t 0 ρ, a ρ ) B by t 0 ρ A B/C and the definition of A B/C. Suppose Γ t 0 : A B/C Γ, x : A t : D Γ, y : C q : D else-el Γ ε 2 (t 0, λx.t, λy.q) : D Then ε 2 (t 0, λx.t, λy.q) ρ = ε 2 ( t 0 ρ, λx. t ρ, λy. q ρ ) D by t 0 ρ A B/C and the definition of A B/C. Suppose Γ a : A Γ b : B then-in Γ ι 1 (a, b) : A B/C Let a A. Then ε 1 (ι 1 ( a, b ρ ), a ) k b ρ B, so ε 1 (ι 1 ( a, b ρ ), a ) B, because ε 1 (ι 1 ( a, b ρ ), a ) SN. Let D SAT and let t and q satisfy a X(t[x := a ] D) and c Z(q[y := c] D). We have ε 2 (ι 1 ( a ρ, b ρ ), λx.t, λy.q) k t[x := a ρ ] D, because ε 2 (ι 1 ( a ρ, b ρ ), λx.t, λy.q) SN. So ι 1 (a, b) ρ = ι 1 ( a ρ, b ρ ) A B/C. Suppose Γ, z : A s : A B/C Γ c : C else-in Γ ι 1 (λz.s, c) : A B/C Let a A. Then ε 1 (ι 1 (λz. s ρ, c ρ ), a ) k ε 1 ( s ρ [z := a ], a ). The induction hypothesis says that s ρ [z := a ] = s ρ[z:=a ] A B/C, so ε 1 ( s ρ [z := a ], a ) B and so ε 1 (ι 1 (λz. s ρ, c ρ ), a ) B, because ε 1 (ι 1 (λz. s ρ, c ρ ), a ) SN Let D SAT and let t and q satisfy a X(t[x := a ] D) and c Z(q[y := c] D). We have ε 2 (ι 1 (λz. s ρ, c ρ ), λx.t, λy.q) k q[y := c ρ ] D, so ε 2 (ι 1 (λz. s ρ, c ρ ), λx.t, λy.q) D, because ε 2 (ι 1 (λz. s ρ, c ρ ), λx.t, λy.q) SN.